T-Stresses for Edge Cracks and Vanishing Ligaments

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
John P. Dempsey
Keyword(s):  
Half Plane ◽  
Edge Crack ◽  
Flexural Rigidity ◽  
Quantitative Relationship ◽  
Crack Mouth ◽  
Crack Face ◽  
T Stress ◽  
Concentrated Loading ◽  
Edge Cracks ◽  
Uncracked Ligament

An edge-cracked half-plane 0 < x < A and a half-plane x > 0 with a semi-infinite crack x > a perpendicular to the edge are examined in this paper. Uniform crack-face loading is thoroughly examined, with a thorough exposition of the Koiter Wiener–Hopf approach (Koiter, 1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374); an analytical expression for the corresponding T-stress is obtained. For the additional cases of (i) nonuniform edge-crack crack-face loading σ(x/A)k (ℜ(k)>-1), (ii) concentrated loading at the edge-crack crack mouth, the Wiener–Hopf solutions and analytical T-stress expressions are provided, and tables of T-stress results for σ(x/A)k and σ(1-x/A)k are presented. A Green's function for the edge-crack T-stress is developed. The differing developments made by Koiter (1956, “On the Flexural Rigidity of a Beam Weakened by Transverse Saw Cuts,” Proc. Royal Neth. Acad. of Sciences, B59, pp. 354–374, Wigglesworth (1957, “Stress Distribution in a Notched Plate,” Mathematika, 4, pp. 76–96), and Stallybrass (1970, “A Crack Perpendicular to an Elastic Half-Plane,” Int. J. Eng. Sci., 8, pp. 351–362) for the case of an edge-cracked half-plane are enhanced by deducing a quantitative relationship between the three different Wiener–Hopf type factorizations. An analytical universal T-stress expression for edge-cracks is derived. Finally, the case of a vanishing uncracked ligament in a half-plane is examined, and the associated Wiener–Hopf solution and analytical T-stress expression are again provided. Several limiting cases are examined.

Weight Functions for T-Stress for Edge Cracks in Thick-Walled Cylinders

2004 ◽  
Author(s):  
J. Li ◽  
C. L. Tan ◽  
X. Wang
Keyword(s):  
Boundary Element ◽  
Edge Crack ◽  
Complex Stress ◽  
Weight Functions ◽  
Stress Distributions ◽  
Internal Edge ◽  
Crack Face ◽  
Nonlinear Stress ◽  
Wide Range ◽  
T Stress

This paper presents T-stress solutions for an internal edge crack in thick-walled cylinders under complex stress distributions. First, the background of the weight function method for the calculation of T-stress is discussed. Then the T-stress results for edge-cracked cylinders obtained from extensive boundary element analyses are summarized. The crack geometries analyzed cover a wide range of radius ratios and relative crack lengths. The loading cases considered in the BEM analysis for the cracked cylinder are: i) crack face pressures with polynomial stress distributions acting on the crack face and ii) internal pressure or steady state thermal loading in the cylinder. Then, the T-stress results for uniform and linearly varying crack face pressure cases are used as the reference solutions to derive weight functions for T-stress. Boundary element results of T-stress for other nonlinear stress distributions are used to validate the derived T-stress weight functions. Excellent accuracy has been achieved. The weight functions derived are suitable for obtaining T-stress solutions for thick-walled cylinders with an internal edge crack under any complex stress fields.

SINGULAR INTEGRAL EQUATION METHOD FOR MULTIPLE CURVED EDGE CRACKS EMANATING FROM BOUNDARY OF HALF-PLANE

2006 ◽  
Vol 03 (02) ◽  
pp. 205-217 ◽  
Author(s):  
Y. Z. CHEN ◽  
X. Y. LIN
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Edge Crack ◽  
Crack Problem ◽  
Infinite Plate ◽  
Complex Potentials ◽  
Edge Cracks ◽  
Edge Crack Problem

This paper provides, an elastic solution for multiple curved edge cracks emanating from the boundary of the half-plane. After placing the distributed dislocations at the prospective sites of cracks in an infinite plate, the principal part of the complex potentials is obtained. By using the concept of the modified complex potentials, the complementary part of the complex potentials can be derived. The whole complex potentials satisfy the traction free condition along the boundary of half-plane automatically. This is a particular advantage of the suggested method. This concept or method of the modified complex potentials is a counterpart of the Green's function method, which is universal in mathematical physics. The direct usage of this method cannot provide a solution in detail. Comparing with the line edge crack case, the following points are significant in the presented study. The relevant kernels in the integral equation are more complicated than in the line edge crack case and the relevant integrations in the problem should be completed on curves. This paper solves a rather complicated problem, the multiple curved edge crack problem, and gives the final solution. A singular integral equation is formulated with the dislocation distribution being unknown function and the traction being the right hand term. The singular integral equation is solved by using the curve length method in conjunction with the semiopening quadrature rule. Periodic curved edge crack problem is also addressed. Finally, several numerical examples are given to illustrate the efficiency of the method presented.

Weight Functions for T-Stress for Edge Cracks in Thick-Walled Cylinders

2005 ◽  
Vol 127 (4) ◽  
pp. 457-463 ◽  
Author(s):  
Jian Li ◽  
Choon-Lai Tan ◽  
Xin Wang
Keyword(s):  
Boundary Element ◽  
Edge Crack ◽  
Thermal Loading ◽  
Complex Stress ◽  
Weight Functions ◽  
Stress Distributions ◽  
Internal Edge ◽  
Crack Face ◽  
Wide Range ◽  
T Stress

This paper presents T-stress solutions for an internal edge crack in thick-walled cylinders. Elastic fracture mechanics analysis using the boundary element method (BEM) is performed to determine the T-stress solutions for a wide range of radius ratios and relative crack lengths. The loading cases considered in the BEM analysis for the cracked cylinder are crack-face pressures with polynomial stress distributions acting on the crack face. T-stress results for the uniform and linearly varying crack-face pressure cases are subsequently used as the reference solutions to derive weight functions for T-stress. Boundary element results of T-stress for other stress distributions, namely, other nonlinear crack face loading, internal pressure, and steady-state thermal loading, are used to validate the derived T-stress weight functions. Excellent agreement between the results from the weight function predictions and those directly computed is shown to be obtained. The weight functions derived are suitable for obtaining T-stress solutions for thick-walled cylinders with an internal edge crack under any complex stress fields.

T-Stress Solutions for Multiple Edge Cracks in Thick-Walled Cylinders

2005 ◽  
Author(s):  
J. Li ◽  
X. Wang ◽  
C. L. Tan
Keyword(s):  
Weight Function ◽  
Boundary Element ◽  
Complex Stress ◽  
Weight Functions ◽  
Stress Distributions ◽  
Internal Edge ◽  
Crack Face ◽  
Nonlinear Stress ◽  
T Stress ◽  
Edge Cracks

In this paper, the boundary element method (BEM) is employed to obtain T-stress solutions for multiple internal edge-cracks in thick-walled cylinders under complex stress distributions. Thick-walled cylinders with two, four and eight internal edge cracks, respectively, are analyzed for a wide range of radius ratios and relative crack lengths. The load cases considered for the cracked cylinder are: i) crack face pressure with uniform, linear, quadratic and cubic stress distributions and ii) internal pressure in the cylinder. The T-stress results for the uniform and the linear distribution crack face pressure cases are used as the reference solutions to derive weight function solutions for the T-stress for the corresponding geometries. The direct boundary element results of the T-stress for the other nonlinear stress distributions are used to validate the derived T-stress weight functions. Excellent agreements between the BEM results and weight function predictions are obtained. The weight functions derived are suitable for obtaining T-stress solutions for the corresponding cracked thick-walled cylinder under any complex stress fields.

Growth of 3D edge cracks in mode I and T-stress on the atomistic level

2017 ◽  
Vol 138 ◽  
pp. 315-322 ◽  
Author(s):  
Anna Machová ◽  
Alena Uhnáková ◽  
Petr Hora
Keyword(s):  
Mode I ◽  
T Stress ◽  
Edge Cracks

Contact interaction of the borders of edge cracks in a flexible half plane

2000 ◽  
Vol 101 (1) ◽  
pp. 2789-2793
Author(s):  
V. V. Perepichka ◽  
I. P. Shatskii
Keyword(s):  
Half Plane ◽  
Contact Interaction ◽  
Edge Cracks

Analysis for the Propagation of Edge Crack of Silicon Steel during Cold Rolling Process Based on GTN Damage Model

2012 ◽  
Vol 482-484 ◽  
pp. 487-492
Author(s):  
Yu Xi Yan ◽  
Quan Sun ◽  
Jian Jun Chen ◽  
Hong Liang Pan
Keyword(s):  
Cold Rolling ◽  
Edge Crack ◽  
Damage Model ◽  
Rolling Process ◽  
Silicon Steel ◽  
Damage Distribution ◽  
Initiation And Propagation ◽  
Gtn Damage Model ◽  
Edge Cracks ◽  
Good Agreement

Silicon steels tend to develop edge cracks during cold rolling, which need to be removed and cause rupture of the steel in the rolling mill. Hence, it is necessary to understand the formation of edge cracks. The damage distribution and the initiation and propagation of edge cracks occur around the notch tip during cold rolling process was investigated by using GTN damage model. The damage parameters f0, fcand fFare determined by tension experiments and SEM observation. The influence of various rolling parameters on damage distribution and crack length was simulated by using ABAQUS. The numerical results show that the GTN damage model is available to prediction the initiation and propagation of edge cracks during rolling process. Parametric study carried out in this present work reveals that the possible occurrence of edge cracks is higher at larger reduction, higher friction coefficient, smaller roll radius and stronger unit tension. The simulation and experimental results have a good agreement .

Certain cases of elastic equilibrium of a composed wedge with an interface crack

2020 ◽  
Vol 25 (12) ◽  
pp. 2199-2209
Author(s):  
Konstantin B Ustinov
Keyword(s):  
Half Plane ◽  
Edge Crack ◽  
Elastic Equilibrium ◽  
Restrictive Condition ◽  
Uniform Loading ◽  
The Matrix ◽  
Remote Loading ◽  
The Common ◽  
Inclined Edge Crack ◽  
Crack Faces

Problems of interface cracks starting from the common corner points of pairs of perfectly glued wedges of different isotropic elastic materials are addressed. It is demonstrated that for a few particular configurations and a restrictive condition imposed on values of elastic constants (corresponding to vanishing of the second Dundurs parameter), the problem of elastic equilibrium may be solved by Khrapkov’s method. These configurations are: (i) the wedges forming a half-plane; (ii) the wedges forming a plane; (iii) one of the wedges being a half-plane. In all cases, the external boundaries are supposed to be free of stresses. By applying Mellin’s transform for all three configurations the problem has been reduced to vector Riemann’s problem, and the matrix coefficient has been factorized for the case of the mentioned restrictive condition. The first configuration, i.e. the problem of an inclined edge crack located along the boundary separating two wedges of different elastic isotropic materials forming a half-plane is considered in more detail. The solution has been obtained for both uniform (corresponding to remote loading) and non-uniform (loading applied at the crack faces) problems. Numerical results are presented and compared with the available results obtained by other authors for particular cases. The obtained solutions appear especially valuable for analysing extreme cases of parameters.

An edge crack in an orthotropic half plane

1987 ◽  
Vol 23 (1) ◽  
pp. 65-69
Author(s):  
V. D. Kuliev ◽  
N. D. Zhekov
Keyword(s):  
Half Plane ◽  
Edge Crack
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